W14 Notes

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title: W14 Notes

Polar curves

01 Theory - Polar points, polar curves

Polar coordinates are pairs of numbers which identify points in the plane in terms of distance to origin and angle from -axis:

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Converting

Polar coordinates have many redundancies: unlike Cartesian which are unique!

For example:
- And therefore also (negative can happen)
For example: for every
For example: for any

Polar coordinates cannot be added: they are not vector components!

For example
Whereas Cartesian coordinates can be added:

⚠️ The transition formulas require careful choice of .

The standard definition of sometimes gives wrong
- This is because it uses the restricted domain ; the polar interpretation is: only points in Quadrant I and Quadrant IV (SAFE QUADRANTS)
Therefore: check signs of and to see which quadrant, maybe need -correction!
- Quadrant I or IV: polar angle is
- polar angle is

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Equations (as well as points) can also be converted to polar.

For , look for cancellation from .

For , try to keep inside of trig functions.

For example: $⨠ ⨠$

02 Illustration

Converting to polar:

-correction

Converting to polar: -correction

Compute the polar coordinates of and of .

Solution

For we observe first that it lies in Quadrant II.

Next compute:

$⨠ ⨠ ⨠ ⨠$

This angle is in Quadrant IV. We add to get the polar angle in Quadrant II:

$⨠ ⨠$

The radius is of course since this point lies on the unit circle. Therefore polar coordinates are .

For we observe first that it lies in Quadrant IV.

Next compute:

$⨠ ⨠ ⨠ ⨠$

This is the correct angle because Quadrant IV is SAFE. So the point in polar is .

Shifted circle in polar

For example, let’s convert a shifted circle to polar. Say we have the Cartesian equation:

Then to find the polar we substitute and and simplify:

$⨠ ⨠ ⨠ ⨠ ⨠ ⨠ ⨠ ⨠ ⨠ ⨠$

So this shifted circle is the polar graph of the polar function .

03 Theory - Polar limaçons

To draw the polar graph of some function, it can help to first draw the Cartesian graph of the function. (In other words, set and , and draw the usual graph.) By tracing through the points on the Cartesian graph, one can visualize the trajectory of the polar graph.

This Cartesian graph may be called a graphing tool for the polar graph.

A limaçon is the polar graph of .

Any limaçon shape can be obtained by adjusting in this function (and rescaling):

Limaçon satisfying : unit circle.

Limaçon satisfying : ‘outer loop’ circle with ‘dimple’:

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Limaçon satisfying : ‘cardioid’ ‘outer loop’ circle with ‘dimple’ that creates a cusp:

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Limaçon satisfying : ‘dimple’ pushes past cusp to create ‘inner loop’:

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Limaçon satisfying : ‘inner loop’ only, no outer loop exists:

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Limaçon satisfying : ‘inner loop’ and ‘outer loop’ and rotated :

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Transitions between limaçon types, :

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Notice the transition points at and :

The flat spot occurs when

Smaller gives convex shape

The cusp occurs when

Smaller gives dimple (assuming )
Larger gives inner loop

04 Theory - Polar roses

Roses are polar graphs of this form:

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The pattern of petals:

(even): obtain petals
- These petals traversed once
(odd): obtain petals
- These petals traversed twice
Either way: total-petal-traversals: always

Calculus with polar curves

05 Theory - Polar tangent lines, arclength

Polar arclength formula

The arclength of the polar graph of , for :

To derive this formula, convert to Cartesian with parameter :

$⨠ ⨠$

From here you can apply the familiar arclength formula with in the place of .

Extra - Derivation of polar arclength formula

Let and convert to parametric Cartesian, so and .

Then:

$⨠ ⨠ ⨠ ⨠$

Therefore:

$⨠ ⨠$

Therefore:

$⨠ ⨠$

Therefore:

06 Illustration

Finding vertical tangents to a limaçon

Let us find the vertical tangents to the limaçon (the cardioid) given by .

& Convert to Cartesian parametric.
- Plug into and : $⨠ ⨠$
&& Compute and .
- Derivatives of both coordinates: $⨠ ⨠$
- Simplify: $⨠ ⨠$
&&& The vertical tangents occur when .
- Set equation: : $⨠ ⨠$
- !! Solve equation.
  - Convert to only : $⨠ ⨠$
  - Substitute and simplify: $⨠ ⨠ ⨠ ⨠$
  - Solve: $⨠ ⨠ ⨠ ⨠$
  - Solve for : $⨠ ⨠ ⨠ ⨠$
&&& Compute final points.
- In polar coordinates, the final points are: $⨠ ⨠$
- In Cartesian coordinates:
  - For : $⨠ ⨠ ⨠ ⨠ ⨠ ⨠$
  - For : $⨠ ⨠ ⨠ ⨠ ⨠ ⨠$
  - For : $⨠ ⨠ ⨠ ⨠ ⨠ ⨠$
!! Correction: is a cusp.
- The point at is on the cardioid, but the curve is not smooth there, this is a cusp.
- Still, the left- and right-sided tangents exists and are equal, so in a sense we can still say the curve has vertical tangent at .

Length of the inner loop

Consider the limaçon given by . How long is its inner loop? Set up an integral for this quantity.

Solution

The inner loop is traced by the moving point when . This can be seen from the graph:
Pasted image 20241124160512.png

Therefore the length of the inner loop is given by this integral:

$⨠ ⨠$

07 Theory - Polar area

Sectorial area from polar curve

The “area under the curve” concept for graphs of functions in Cartesian coordinates translates to a “sectorial area” concept for polar graphs. To compute this area using an integral, we divide the region into Riemann sums of small sector slices.

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To obtain a formula for the whole area, we need a formula for the area of each sector slice.

Area of sector slice

Let us verify that the area of a sector slice is .

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Take the angle in radians and divide by to get the fraction of the whole disk.

Then multiply this fraction by (whole disk area) to get the area of the sector slice.

$⨠ ⨠$

Now use and for an infinitesimal sector slice, and integrate these to get the total area formula:

One easily verifies this formula for a circle.

Let be a constant. Then:

$⨠ ⨠ ⨠ ⨠$

The sectorial area between curves: